Answer: hello!
Assuming you don't know much maths, so no quadratic identities, no first and second derivatives, no completing the square:
Note if you graph y=x and y=4−x , that the lines intersect (cross) at x=2 .
Also note that if you move right or left on the x axis, whatever you add to x , you subtract from 4−x .
So for example, if x=2.1=2+0.1 then 4−x=4−2.1=1.9=2−0.1
Therefore we can rewrite x(x−4) as (2+a)(2−a)=4−a2 taking as how far we are from x=2 . In the above example, a=0.1.
Since a positive times a positive is positive, and a negative times a negative is positive, then a2≥0 , and clearly 4−a2 is greatest at a=0 , which is the same as saying that x(4−x) is maximum at x=2 .
<h3>also:</h3><h3 />
x(4−x)=4x−x2=−x2+4x
This is a quadratic function (y=ax2+bx+c) . The x -vertex is
xv=−b2a
xv=−42⋅(−1)=−4−2
xv=2
We can solve this problem with derivatives too.
y=x(4−x)
y′=[x(4−x)]′
With Product Rule, we have:
y′=(1)(4−x)+x(−1)=4−x−x
The maximum (or minimum) is a zero of the derivative of y . So, we have
4−2x=0
4=2x
42=x
2=x
x=2
